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Complex Numbers
Difficulty: Hard
Question Paper 1
Level
A Level
Subject
Maths Pure 3
Exam Board
CIE
Topic
Complex Numbers
Difficulty
Hard
Booklet
Question Paper 1
Time allowed:
76 minutes
Score:
/54
Percentage:
/100
Grade Boundaries:
1
A*
A
B
C
D
E
>90%
81%
70%
58%
46%
34%
Question 1
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(a) The equation 2x 3 − x2+ 2 x + 12 = 0 has one real root and two complex roots. Showing your
working, verify that 1 + i √3 is one of the complex roots. State the other complex root.
[4]
√
3.
(b) On a sketch of an Argand diagram, show the point representing the complex number 1 + i
On the same diagram, shade the region whose points represent the complex numbers z which
1
[5]
satisfy both the inequalities |z −1 − i √3 | ≤ 1 and arg z ≤ .
3
2
Question 2
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(a) It is given that −1 + (√5)i is a root of the equation z3+ 2z + a = 0, where a is real. Showing your
working, find the value of a, and write down the other complex root of this equation.
[4]
(b) The complex number w has modulus 1 and argument 28 radians. Show that
3
w−1
= i tan . [4]
w +1
Question 3
7
(a) The complex number
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3 − 5i
is denoted by u. Showing your working, express u in the form
1 + 4i
x + iy, where x and y are real.
[3]
(b)
(i) On a sketch of an Argand diagram, shade the region whose points represent complex
[4]
numbers satisfying the inequalities
(ii) Calculate the maximum value of arg z for points lying in the shaded region.
4
[2]
Question 4
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The complex number w is defined by w =
22 + 4i
.
(2 − i)2
(i) Without using a calculator, show that w = 2 + 4i.
(ii) It is given that p is a real number such that
arg of p.
1
4
≤
[3]
(w + p) ≤ 43 . Find the set of possible values
[3]
(iii) The complex conjugate of w is denoted by w* . The complex numbers w and w * are represented
in an Argand diagram by the points S and T respectively. Find, in the form lz − al = k, the
equation of the circle passing through S, T and the origin.
[3]
5
Question 5
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The complex number u is given by u = −1 + (4√3)i.
(i) Without using a calculator and showing all your working, find the two square roots of u. Give
your answers in the form a + ib, where the real numbers a and b are exact.
[5]
(ii) On an Argand diagram, sketch the locus of points representing complex numbers z satisfying
the relation lz − u l = 1. Determine the greatest value of arg z for points on this locus.
[4]
6
Question 6
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Throughout this question the use of a calculator is not permitted.
(a) Solve the equation (1 + 2i) w 2 + 4w − (1 − 2i) = 0, giving your answers in the form x + iy, where
x and y are real.
[5]
(b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers
satisfying the inequalities lz − 1 − il ≤ 2 and −14  ≤ arg z ≤14 .
7
[5]